フーリエ・ルジャンドル法の誤差特性

Abstract

Error characteristics of the Fourier expansion of the Legendre polynomials are examined in the computation of the Gaussian latitudes and weights and in the Fourier– Legendre scaling factor for associated Legendre functions. The Fourier–Newton method yields values accurate enough for practical use up to 10000 nodes. Because the relative error grows in O(𝑛) and the computational complexity in O(𝑛"), alternative methods may be required for larger calculation. There is a room for improvement in the computation of the Fourier–Legendre coefficients. The Fourier–Legendre scaling factor is not necessary in the computation of the Gaussian latitudes and weights, but for associated Legendre functions. The accuracy can be improved significantly by use of alternative equations or the asymptotic series of the Gamma function.